Stated in simplest terms, this article considers, in an abstract mathematical framework, a curve fitting or estimation problem where a given set of data points f is approximated or estimated by an element from a set K so that the estimate of f is least affected by perturbations in f.
Let X be a normed linear space with norm ∥·∥ and K be any (not necessarily convex) nonempty subset of X. For any f in X, let
denote the shortest distance from f to K. Let also, for f in X,
The set-valued mapping P on X is called the metric projection onto K. It is also called the nearest point mapping , best approximation operator , proximity map , etc. If P(f) ≠ø, then each element in it is called a best approximation to (or a best estimate of) f from K. In practical curve fitting or estimation problems, f represents the given data and the set K is dictated by the underlying process that generates f. Because of random disturbance or noise, f is in general not in K, and it is required to estimate f...
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Ubhaya, V.A. (2001). Best Approximation by Bounded or Continuous Functions . In: Floudas, C.A., Pardalos, P.M. (eds) Encyclopedia of Optimization. Springer, Boston, MA. https://doi.org/10.1007/0-306-48332-7_25
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DOI: https://doi.org/10.1007/0-306-48332-7_25
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